GOAL Examine whether a transformation is linear Find the ima

GOAL Examine whether a transformation is linear. Find the image and kernel of a linear transformation. Examine whether a linear transformation is an isomorphism. Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = M + I_2 from R^2 Times 2 to R^2 Times 2 2 T (M) = 7M from R^2 Times 2 to R^2 Times 2 T(M) = (sum of the diagonal entries of M) from R^2 Times 2 to R T(M) = det(M) from R^2 Times 2 to R T(M) = M[1 2 3 6] from R^2 Times 2 to R^2 Times 2 T (M) = [1 2 3 4] M from R^2 Times 2 to R^2 Times 2 T(M) = M[1 2 3 4] M from R^2 Times 2 to R^2 Times 2 T(M) = S^-1 MS, where S = [3 4 5 6], from R^2 Times 2 to R^2 Times 2 T(M) = P M P^-1 MS, where P = [2 3 5 7], from R^2 Times 2 to R^2 Times 2 T(M) = PMQ, where P = [2 3 5 7] and Q =

Solution

An isomorphism is a homomorphism that admits an inverse. A homomorphism is a structure-preserving map between two algebraic structures i.e. a transformation of one set into another that preserves in the second set the relations between elements of the first.

2. We have T: R^2x2 R^2x2 such that T (M) = 7M.

Let M and N be two arbitrary elements of R^2x2   and let be an arbitrary scalar. Then T(M + N) = 7(M+N) = 7M + 7N = T(M) + T(N) . Thus T is closed under addition. Further, T ( M) = 7( M) = (&M) = T(M). Therefore, T is closed under scalar multiplication also and hence T is a linear transformation.

Im (T) = { T(M) : M R^2x2 } = { 7M : M R^2x2 } ; Ker(T) = { M R^2x2 : T(M) = 0 } = { M R^2x2 : 7M = 0 } = {0}. Thus Ket (T) is the 2 x 2 zero matrix.

Further, T (M) = 7M so that T(1/7M) = M and hence T^-1 (M) = 1/7 M . Thus T is invertible and hence T is an isomorphism.

4. T(M) = det (M)

Let M and N be two arbitrary elements of R^2x2 . Then T (M+ N) = det (M + N) det (M) + Det (N) . Thus,         T (M+N) T(M) + T(N). Therefore, T is not closed under addition and, therefore, T is not a linear transformation.

8.T(M)= [ 1 2 ]    M = AM ( say) where, A = [ 1 2 ]

[ 3 4 ] [ 3 4 ]

Then T (M + N) = A(M+N) = AM + AN =T(M) + T(N) . Thus T is closed under addition. Also, T ( M) = A ( M) = (A M) = T(M) . Thus T is also closed under scalar multiplication and therefore, T is a linear transformation.

Im (T) = { T(M) : M R^2x2 } = { AM : M R^2x2 } ; Ker(T) = { M R^2x2 : T(M) = 0 } = { M R^2x2 : AM = 0 } = {0}. As A is not a zero matrix.

Further, T(M) = AM so that T(A^-1 M) = A^-1 (AM) = (A^-1A)M = IM = M . Also A^-1 = [ -2     1 ]

[3/2 - ½ ]

Thus T^-1 (M) = A^-1M and thus T is invertible and therefore, an isomorphism.